Preheat your oven to 325°F (165°C). Grease and flour a 9x9 inch (23x23 cm) square baking pan.
In a large, clean bowl, use an electric mixer to beat the egg whites on high speed until stiff peaks form. This means when you lift the beaters, the peak holds its shape. Set the bowl aside.
In a separate large bowl, beat the egg yolks, granulated sugar, and peppermint extract on high speed for about 2 minutes, until the mixture is pale yellow and creamy.
In a microwave-safe bowl, combine the cubed butter and chopped white chocolate. Microwave at 50% power in 30-second intervals, stirring each time, until everything is melted and smooth.
Pour the melted white chocolate mixture into the egg yolk mixture and mix until just combined. Add the flour and salt, and mix again on low speed until no dry spots remain.
With the mixer still on low, slowly pour in the warm milk. Continue mixing until the batter is smooth. It will be very thin, similar to crepe batter.
Gently fold about one-third of the whipped egg whites into the batter using a spatula to lighten it. Add the rest of the egg whites and continue to fold gently until just combined. A few small lumps are okay.
Pour the batter into your prepared pan. Bake for 50-55 minutes. The top will be golden brown, and the cake should be cooked through. For accuracy, an instant-read thermometer inserted into the center should read 160°F (71°C).
Place the pan on a wire rack and let the cake cool completely to room temperature. Then, cover it and refrigerate for at least 4 hours, or overnight. This chilling step is essential. Serve cold, topped with whipped cream and crushed peppermints.